A question of long-standing interest to philosophers, psychologists and neuroscientists is how the brain selects which signals enter consciousness. Binocular rivalry and attention both involve selection of visual stimuli, but affect perception quite differently. During binocular rivalry, awareness alternates between two different stimuli presented to the two eyes. In contrast, attending to one of two different stimuli impairs discrimination of the ignored stimulus, but without causing it to disappear from consciousness. Here we show that despite this difference, attention and rivalry rely on shared object-based selection mechanisms. We cued attention to one of two superimposed transparent surfaces and then deleted the image of one surface from each eye, resulting in rivalry. Observers usually reported seeing only the cued surface. They were also less accurate in judging unpredictable changes in the features of the uncued surface. Our design ensured that selection of the cued surface could not have resulted from spatial, ocular or feature-based mechanisms. Rather, attention was drawn to one surface, and this caused the other surface to be perceptually suppressed during rivalry. These results raise the question of how object representations compete during these two forms of perceptual selection, even as the features of those objects change unpredictably over time.